Problem: Solve for $x$, ignoring any extraneous solutions: $\dfrac{x^2 - 11}{x + 2} = \dfrac{-11x - 29}{x + 2}$
Answer: Multiply both sides by $x + 2$ $ \dfrac{x^2 - 11}{x + 2} (x + 2) = \dfrac{-11x - 29}{x + 2} (x + 2)$ $ x^2 - 11 = -11x - 29$ Subtract $-11x - 29$ from both sides: $ x^2 - 11 - (-11x - 29) = -11x - 29 - (-11x - 29)$ $ x^2 - 11 + 11x + 29 = 0$ $ x^2 + 18 + 11x = 0$ Factor the expression: $ (x + 9)(x + 2) = 0$ Therefore $x = -9$ or $x = -2$ However, the original expression is undefined when $x = -2$. Therefore, the only solution is $x = -9$.